Louis Nirenberg: 2015 Abel Prize for His Contributions to the Theory of Pdes

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Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs

Abel Prize 2015 to the American mathematicians John F. Nash, Jr. and Louis Nirenberg ''for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis''

Xavier Cabré ICREA Research Professor at the UPC 06/05/2015 FME-UPC Louis Nirenberg:

● Born Feb. 28, 1925 in Hamilton, Ontario, Canada, in a Jewish family

● Master and Graduate School at New York University

● Ph.D. 1949 under the direction of James Stoker

● Since then, Faculty at the Courant Institute of Mathematical Sciences, New York University. He retired 1999

L. Nirenberg: B.Sc. from McGill University (Canada), 1945.

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Richard Courant

at the Courant Institute of Mathematical Sciences (New York University)

Founded: 1935 Current Building: 1965

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Courant Institute Directors after Richard Courant: http://cims.nyu.edu/webapps/cont ent/about/history http://www.cims.nyu.edu/gallery/

Kurt O. Friedrichs

L. Nirenberg (left) and Jürgen Moser Peter Lax (right; Abel Prize 2005) S.R. Snirivasa Varadhan (Abel Prize 2007)

Allyn Jackson, 2002, “Interview with Louis Nirenberg”

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré James J. Stoker (PhD advisor) and Louis Nirenberg

Kurt O. Friedrichs (left) and Richard Courant (right)

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré "Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré "Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré L. Nirenberg's main awards and honors:

● the American Mathematical Society’s Bôcher Prize in 1959

● the Jeffrey-Williams Prize of the Canadian Mathematical Society in 1987

● the Steele Prize of the AMS in 1994 for Lifetime Achievement

● First recipient in mathematics of the Crafoord Prize, in 1982, established by the Royal Swedish Academy of Sciences in areas not covered by the Nobel Prizes. He shared the award with Vladimir Arnold

● Inaugural Chern Medal, in 2010, given by the International Mathematical Union and the Chern Medal Foundation

● Abel Prize 2015, with John F. Nash, Jr.

Receiving the Crafoord Prize, Stockholm 1982, with Mrs. Crafoord and the King of Sweden

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré "Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Some important mathematical contributions of L. Nirenberg:

● Differential Geometry: The Weyl problem; the Minkowski problem

● Complex Analysis: Newlander-Nirenberg theorem; complex Monge-Ampère equations

● Real Analysis: The BMO space; degree of VMO maps

● Theory of PDEs: Regularity theory (inequalities and estimates); Monge-Ampère and fully nonlinear equations; free boundaries; Navier-Stokes equations; symmetry theorems (the moving planes and sliding methods); maximum principles; front propagation; etc, etc.

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Partial Differential Equations. Types : 1. Elliptic : Laplace equation: 2. Parabolic :

● Heat or diffusion equation:

● Navier-Stokes (or 1 million $) equations (incompressible viscous fluids) 3. Hyperbolic :

● Wave equation (acoustics, sound-waves)

● Schrödinger equation (quantum mechanics)

● Euler's equations (incompressible fluids)

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Nirenberg's PhD Thesis (published in the above paper) solves an Open Problem of H. Weyl from 1916: Given a smooth metric g of positive curvature on the sphere S², is there an embedding X: S² R³ such that the metric induced on S² by this embedding is g?

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Soap films = Minimal surfaces

Dirichlet integral

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Soap films = Minimal surfaces

Dirichlet integral

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré The Laplacian coming from physics: heat, concentrations, gravitational and electrical potentials, viscous fluids, etc.

Fourier law for heat

The heat equation Stationary solutions

Pierre-Simon, Jean-Baptiste marquis de Laplace Joseph Fourier (1749-1827) (1768-1830)

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré The Laplacian coming from Finance and Probability: what is your expected gain when, starting always from the same given tile in your living room, you walk randomly and you get 30€ only when you hit a radiator on the first time that you hit your living room's walls (otherwise you get 0€)?

0€ 30€

0€

30€

30€ 0€

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré How to solve the problem:

● make a squared lattice of very small step-size h

● Move from a point to either East, West, North, or South,

each one with probability 1/4 N

W E 0€ 30€ C

0€ S C = starting point of the walk

0€ u(C) = expected gain starting from C

30€

(average)

30€ 0€

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré

h = step size of the lattice

The LAPLACIAN of u = 0

● X. Cabré, Partial differential equations, geometry and stochastic control, in Catalan. Butl. Soc. Catalana Mat. 15 (2000), 7-27 ● X. Cabré, Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Existence theorems come from estimates, which give compactness:

Poincaré inequality:

Rellich–Kondrachov theorem, an extension of

Arzelà-Ascoli theorem:

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Poincaré inequality:

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré + initial (and boundary) conditions

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Sobolev-Gagliardo-Nirenberg inequalities (u=0 on boundary of B_r):

if n=2

if n=3

Area functional of films, graphs:

Minimal surface equation for graphs:

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Back to Geometry:

K(x) = scalar curvature u(x) = conformal factor The Yamabe problem

K(x) = Gauss curvature Surface = graph of u

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré solves an Open Problem of H. Weyl from 1916: Given a smooth metric g of positive curvature on the sphere S², is there an embedding X: S² R³ such that the metric induced on S² by this embedding is g?

● Continuity method and IFT (implicit function theorem)

● Need estimates (regularity) for solutions of Monge- Ampère type equations in dimension 2. This gives that the linearized problem is an isopmorphism (IFT ok) but in HÖLDER or SOBOLEV spaces, NOT from C² to C⁰ :

with zero Dirichlet boundary conditions is not an isomorphism if n > 1

● Continuity method and IFT (implicit function theorem)

● Need estimates (regularity) for solutions of Monge-Ampère type equations in dimension 2. This gives that the linearized problem is an isopmorphism (IFT ok)

● Similarity with Perelman's proof of the Poincaré conjecture: Homotopy with a nonlinear geometric heat equation: the Ricci flow + estimates for analysis of singularities

● Estimates easier in dim 2 (complex variables, harmonic and analytic functions, quasiconformal mappings): work of C. B. Morrey

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Towards estimates and regularity: differentiating the equation (or making difference quotients: Nirenberg's method)

● Quasilinear equations:

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré De Giorgi-Nash-Moser Theorem: Hölder regularity of solutions of

with a_{ij} uniformly elliptic (positive definite matrices) but only bounded and measurable as a function of x in R^n.

● Nash, J. Parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 754-758.

● Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931-954. ''A gold mine'', in Nirenberg's words.

Nash work retaken and presented in:

● Fabes, E. B.; Stroock, D. W. A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338.

Independently proved by:

● De Giorgi, Ennio. Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 1957 25– 43.

and later a new (third) proof by Jürgen Moser

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Complex Analysis:

● A. Newlander and L. Nirenberg, Complex analytic coordinates in almost-complex manifolds, Ann. Of Math. 65 (1957), 391–404

solves the problem on integrability of almost complex structures. It was suggested to Nirenberg by A. Weil and S.S. Chern. When can one reduce a given system of n first order linear PDEs in R^2n to the Cauchy-Riemann equations in C^n, after a smooth change of variables.

Also an extension to a complex form of the classical Frobenius theorem on differential forms.

Foundation stone for:

● K. Kodaira, L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math. 68 (1958), 450–459

establishes the existence of deformations of complex structures on complex manifolds

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Real Analysis: The BMO space of John-Nirenberg

Fritz John (Courant Institute)

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Real Analysis: The BMO space of The Hardy space H¹ : John-Nirenberg ● H¹ * = BMO

● VMO* = H¹

● Brezis- Fritz John (Courant Institute) Nirenberg: degree theory of VMO maps

C^{2,\alpha} estimate up to the boundary for solutions of the Monge-Ampère problem

● Independently proved ny N.V. Krylov (1983)

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré sequence of papers that establish a long time open

C^{2,\alpha} estimate up to the boundary for solutions of the Monge-Ampère problem

● Independently proved ny N.V. Krylov (1983)

Nirenberg with Irene Gamba Joel Spruck and Luis Caffarelli

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Related to the study of bubbles in the Yamabe problem: K(x) = scalar curvature for a new conformal metric u(x) = conformal factor

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré

Quoting L. Nirenberg:

''I made a living off the maximum principle''

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré If u and v are solutions of the same elliptic nonlinear equation, then u - v satisfies a linear elliptic equation with variable coefficients:

Thus, u and v (solutions of the same nonlinear problem) such that u \leq v and u(x_0)=v(x_0), then u \equiv v Two different soap films or soap bubbles cannot touch tangently

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré Proved with the moving planes method of A.D. Alexandrov, who created it to prove: every embedded connected hypersurface in R^n with constant mean curvature must be a ball.

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré ● The sliding method of Berestycki-Nirenberg. Leads to uniqueness, monotonicity of solutions, and Liouville type theorems

Henri Berestycki (Paris 6; now EHESS)

● The maximum principle on non smooth domains:

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré References:

● https://www.simonsfoundation.org/science_lives_video/louis-nirenberg/

(video of J. Shatah's interview to Louis Nirenberg)

● http://www.cims.nyu.edu/gallery/

● http://cims.nyu.edu/webapps/content/about/history

● Allyn Jackson, April 2002, “Interview with Louis Nirenberg,” Notices of the AMS, 49 (4), 441-449

● Simon Donaldson, March 2011, “On the Work of Louis Nirenberg,” Notices of the American Mathematical Society, 58 (3), 469-472

● Yan Yan Li, 2010, “Work of Louis Nirenberg,” Proceedings of the International Congress ofMathematicians, Hyderabad, India

● http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation

(video by Luis Caffarelli's talk on Navier-Stokes)

● X. Cabré, Partial differential equations, geometry and stochastic control, in Catalan. Butl. Soc. Catalana Mat. 15 (2000), 7-27

● X. Cabré, Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457

"Louis Nirenberg: 2015 Abel Prize for his contributions to the theory of PDEs” Xavier Cabré